Two Republican senators suggested Sunday they would not support fellow party member and former Sen. Chuck Hagel should he get nominated for secretary of defense, while Senate Republican Leader Mitch McConnell said he is reserving judgment.

Texas Sen. John Cornyn had the most strident remarks, saying President Obama nominating Hagel sends “the worst possible message” to Israel and other U.S. allies in the Middle East.

“I will not support Chuck Hagel’s nomination,” Cornyn said in a release, citing in part Hagel’s views on Israel and his opposition to Iranian sanctions.

However, he did not say whether he would try to block Hagel’s potential nomination during a Senate confirmation hearing.

A congressional aide said Senate Democrats have been given a heads-up from the White House that Hagel's selection is imminent. And an administration official told Fox News the announcement will be made Monday.

Newly-elected Texas Sen. Ted Cruz told “Fox News Sunday” that “it is very difficult to imagine the circumstance in which I could support his confirmation.”

Cruz also cited Hagel’s views and statements about Israel and Iran and questioned why Obama might pick Hagel in the face of tough opposition.

“The president seems bound and determined to proceed down this path despite the fact that Hagel's record is very, very troubling on the nation of Israel,” Cruz told Fox. “I do think it is interesting that the president seems hell-bent on nominating him despite the fact that a number of prominent Republicans have criticized him and the Democratic senators have been surprisingly silent on it.”

Hagel said in 2006 that "the Jewish lobby intimidates a lot of people" on Capitol Hill, then added, "I'm not an Israeli senator, I'm a United States senator."

McConnell, R-Ky., said any defense nominee must have "a full understanding of our close relationship with our Israeli allies, the Iranian threat and the importance of having a robust military."

McConnell told ABC's "This Week" that Hagel, a Nebraska Republican who left the Senate in 2009, has "certainly been outspoken in foreign policy and defense over the years. The question we will be answering, if he's the nominee, is do his views make sense for that particular job?"

McConnell said he would "wait and see how the hearings go and see whether Chuck's views square with the job he would be nominated to do."

He added: "I'm going to take a look at all the things that Chuck has said over the years and review that, and in terms of his qualifications to lead our nation's military."

Illinois Sen. Dick Durbin, the second-ranking Senate Democrat, told CNN's "State of the Union" that Hagel "is a serious candidate if the president chooses to name him."

Hagel has criticized discussion of a military strike by either the U.S. or Israel against Iran. He also has backed efforts to bring Iran to the table for talks on future peace in Afghanistan.

"This is a controversial pick," Sen. Lindsey Graham, R-S.C., told CNN. "He is an antagonistic figure when it comes to the state of Israel. It's a signal you're sending to Iran at the worst possible time and to our allies."

Officials close to the White House have talked up Hagel as a patriot who served in the military with distinction during Vietnam and someone who shares the president's skepticism about potential conflict with Iran and a large U.S. footprint in Afghanistan.

Liberals have expressed anger over an anti-gay remark Hagel made in the Clinton administration for which he recently apologized.

Fox News' Ed Henry and the Associated Press contributed to this report.